I'm reviewing my undergraduate quantum mechanics from a while back, and I am not quite sure I understand this correctly. I seem to recall being taught that quantization arises from the imposition of boundary conditions (in the mathematical sense). But this isn't quite the same as saying that a particle is in a bound state. I know that particles being bound is related to their total energy vs. the potential they're exposed to, but I'm feeling fuzzy on how these are related. Can anyone help to clarify? Specifically, it's extremely clear to me why energy is quantized in a system like the particle in the box, or even a delta function potential. But it's not intuitively clear to me why energy is quantized for the hydrogen atom - I know that it is, and also that the electron is in a bound state, but I can't *see* this boundedness in the same way that I can for the particle in the box... the mathematics is clear, but I can't think of a great argument in terms of pure physics.